Solve the differential equation (x2 + y2)dx- 2xydy = 0 - Mathematics and Statistics

Question

Solve the differential equation (x² + y²)dx- 2xydy = 0

Sum

Solution

(x² + y²)dx- 2xydy = 0

(x²+ y²) dx = 2xydy

`dy/dx = (x^2 + y^2)/(2xy)`.........(i)

The equation is a homogeneous equation
Let y= vx,
Differentiat ing w.r.t. x, we get,

`dy/dx=v+x(dv)/dx`

`dy/dx=(x^2+y^2)/(2xy) " from "(i)`

`v+x(dv)/dx=(x^2+(vx)^2)/(2x.(vx))`

`v+x(dv)/dx=(1+v^2)/(2v)`

`x(dv)/dx=(1+v^2)/(2v)-v`

`x(dv)/dx=(1+v^2-2v^2)/(2v)`

`x(dv)/dx=(1-v^2)/(2v)`

`(2v)/(1-v^2)dv=1/xdx`.......(ii)

Which is in variables separatable form

∴ Integrating both sides, we get

`int(2v)/(1-v^2)dv=int1/xdx + c_1`

`therefore -log|1-v^2|=log|x|+logc`

`therefore log|x(1-v^2)|=log|c|`

`therefore x(1-v^2)=c`

Resubstituting `v=y/x` we get

`x(1-y^2/x^2)=c`

`x((x^2-y^2)/x^2)=c `

`therefore x^2 - y^2 = cx`, where c is constant

which is the required general solution

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Methods of Solving First Order, First Degree Differential Equations - Homogeneous Differential Equations

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Q 6.2.2 Q 6.2.1 Q 6.2.3

2014-2015 (March)

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2014-2015 (March) (with solutions)

Q 6.2.2 | 4 marks

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